The Annals of Probability

The circular law for random matrices

Friedrich Götze and Alexander Tikhomirov

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We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices.

The results are based on previous work of Bai, Rudelson and the authors extending those results to a larger class of sparse matrices.

Article information

Ann. Probab., Volume 38, Number 4 (2010), 1444-1491.

First available in Project Euclid: 8 July 2010

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Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Circular law random matrices


Götze, Friedrich; Tikhomirov, Alexander. The circular law for random matrices. Ann. Probab. 38 (2010), no. 4, 1444--1491. doi:10.1214/09-AOP522.

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